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Chapter 5: Problem 67

What does it mean if a system of linear inequalities has no solution?

Short Answer

Expert verified
It means that there are no points that satisfy all inequalities in the system at once. This happens when the regions defined by the inequalities do not overlap.

Step by step solution

01

Understanding Linear Inequalities

An inequality is like an equation, but instead of an equals sign, it has a sign showing if something is less than or equal to (\( \leq \)), less than (<), greater than (>), or greater than or equal to (\( \geq \)). When these inequalities are linear, they are called Linear Inequalities. A system of linear inequalities is a set of two or more inequalities with the same variables.
02

Graphical Interpretation

Each linear inequality divides the coordinate plane into two half-planes: one that satisfies the inequality, and one that does not. The solution to the system of inequalities is the region where the half-planes of all inequalities in the system overlap. Each point in this region is a solution to the system.
03

Condition for No Solution

If a system of linear inequalities has no solution, it means that there is no area of overlap between the half-planes defined by the inequalities. Therefore, there is no point that satisfied all inequalities at once.

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Most popular questions from this chapter

Exercises \(47-50\) describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\) c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{array}{l} x=4 y-2 \\ x=6 y+8 \end{array} $$

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is \(7 .\) If one number is subtracted from the other, their difference is \(-1 .\) Find the numbers.

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} (2,5) & \\ 2 x+3 y &=17 \\ x+4 y &=16 \end{aligned} $$

In Exercises \(19-30,\) solve each system by the addition method. \(x+y=1\) \(x-y=3\)
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